After watching this video lesson, you will be able to write the equation of a parabola in standard form when given just two important points from the parabola. Learn what these two points are and how they relate to the parabola.
We can easily define a parabola as an arc. Why are we learning about parabolas? We are learning about parabolas and their standard form equations because these equations are actually used in real life. If it weren't for parabolas, we wouldn't have cell phone service wherever we go. If it weren't for parabolas, we wouldn't have dish television! Cell phones and satellite television both make use of satellites in outer space that are parabola-shaped. The shape of the parabola is the shape you get when you take an ice cream waffle cone and cut it parallel to the side of the cone.
Besides how useful parabolas are, there is one other defining mark of a parabola. It is this defining mark that allows the parabola to magnify bunches of tiny signals to become one big clear signal. What is this? It is the point called the focus on a parabola. This point is located inside the curve of our parabola. It is the point where the distance from any point on the parabola to this focus point is the same as the distance from the point on the parabola to a straight line outside the parabola. This straight line outside the parabola is called the directrix. Why is the focus called the focus? Because of the way they are shaped and because of the property of the focus, parabolas are able to take signals and then reflect all of them to a single focus point so that they become one big clear signal. Yes, parabolas are able to focus signals to a single point.
Of course we would have an equation to describe our parabola algebraically. After all, this is an algebra lesson! We actually have two different forms of our standard equation depending on the direction of our parabola. Our equation is of course based on the Cartesian coordinate plane with the x and y axes. The two forms are dependent on the direction of the parabola.
For parabolas that open either up or down, the standard form equation is (x - h)^2 = 4p(y - k). For parabolas that open sideways, the standard form equation is (y - k)^2 = 4p(x - h). They are essentially the same except that the x and y are switched. Notice also that the h is always with the x and the k is always with the y. One way you can remember this is just like x comes before y in our alphabet, so h comes before k. The 4p then is always with the part that is not squared. The letters h, k and p stand for numbers that give us useful information about the location of our parabola. The vertex or tip of our parabola is given by the point (h, k). If we combine all three letters, it gives us the location of our focus point. For parabolas that open up and down, the focus point is given by (h, k + p). For parabolas that open sideways, the focus point is (h + p, k). Notice how we are adding our p value to the part that tells us which axis the parabola is opening on. For parabolas that open up and down, they open on the y-axis, and so the p value is added to the y value of our vertex. Likewise, sideways parabolas open on the x-axis, and so the p value is then added to the x value of our vertex.
Now that we know what the standard form equation looks like and what the letters stand for, we can now go on and see how we can write a standard form equation. We need to be able to plug in numbers for h, k and p. In order to write our standard form equation, we just need two bits of information about our parabola. We need to know the location of the vertex and the location of our focus. If we know these two things, we are good to go. Let's look at an example.
Let's say our problem tells us that our vertex is located at the point (1, 2) and that our focus is at the point (1, 4). We are to write the standard form equation of our parabola. What do we do first?
First, we will look at our vertex point, and we can immediately label our h and k. We remember that h always comes before k in the alphabet, just like x comes before y, so that means our h is 1 and our k is 2. That was easy. We already have two of the letters we need in order to write our equation.
We just need one more letter, the p. How do we figure out what this letter equals? We look at the location of the focus and compare it to the location of the vertex. In our problem, the focus is located at (1, 4), and the vertex is at (1, 2). We see that the only difference between these two points is the y value. This means that our parabola will open either up or down, so we will use the standard form that begins with x. In our case, because the focus is located above the vertex, our parabola is opening up. Now we need to figure out what the difference is between these points. We need to subtract our vertex point from our focus point, focus point minus vertex point. Our focus point is (1, 4), and our vertex point is (1, 2). So, subtracting, we get (1, 4) - (1, 2) = (0, 2). This means that our p value is 2 on the y-axis.
Now we have all the letters we need to write our standard form equation. We use the one that begins with the x part because our parabola is opening up. We plug in our numbers where they belong. Plug in 1 for h, 2 for k and 2 for p. Our equation is (x - 1)^2 = 4(2)(y - 2). Multiplying the 4 and the 2 together, our final equation is (x - 1)^2 = 8(y - 2), and we are done! Now remember, if we are plugging in a negative number for h or k, our equation will show them as plus because our equation begins with a negative for those letters. You combine a negative with a negative and you end up with a positive.
Now let's review what we've learned. We can say that a parabola is an arc. It has a focus point where the distance from any point on the parabola to this focus point is the same as the distance from the point on the parabola to a straight line outside the parabola. This straight line outside the parabola is called the directrix.
For parabolas that open either up or down, the standard form equation is (x - h)^2 = 4p(y - k). For parabolas that open sideways, the standard form equation is (y - k)^2 = 4p(x - h). The vertex or tip of our parabola is given by the point (h, k). For parabolas that open up and down, the focus point is given by (h, k + p). For parabolas that open sideways, the focus point is (h + p, k).
To write our equation, we need to know the location of the vertex and the focus. The vertex location gives us our h and k. To find our p we take our focus point and subtract the vertex point. If our parabola is opening either up or down, we use the standard form that begins with the x, and if our parabola opens sideways, we use the standard form that begins with y.
Once you have completed this lesson, you should be able to:
- Write the standard form equation for a vertical or horizontal parabola
- Calculate the vertex and the focus point of a parabola